In this section, we derive the approximate formula of the \(v_{Dirac}\)  in the presence of the pseudo-vector field that is induced by the lattice relaxation proposed in Ref. [1]}. Motivated by Fig. REF , we neglect \(\nabla \varphi ^{\mathcal {A}}\) , the irrotational part of the pseudo-vector field \(\mathcal {A}\) , and thus \(\mathcal {A} \approx \nabla \times (\hat{z} \varepsilon ^{\mathcal {A}})\) . Furthermore, as demonstrated in Table REF , the lattice relaxation \(\delta U\)  in Ref. [1]} is dominated by the lowest harmonics. Because \(\varepsilon _{\mathcal {A}}\)  is even and real, its Fourier components must be pure imaginary and also odd. Considering \(C_3\)  symmetry, we can assume that
\(\varepsilon ^{\mathcal {A}}(x) \approx i \tilde{\varepsilon }^{\mathcal {A}}_1 \sum _{j = 1}^3 \left( e^{i g_j \cdot x} - e^{-i g_j \cdot x} \right)\) 
